The proof of Theorem 5.1 in [1]} shows that the assumptions that \(P^{uv}\)  is invertible and not a permutation matrix for every edge \(\lbrace u,v\rbrace \)  must be satisfied for a model \(\theta \in \overline{\Theta }\)  if \(\mathcal {P}_\theta =\mathcal {P}_{\theta _0}\)  for some \(\theta _0\in \Theta \) , by virtue of the form of \(\mathcal {P}_{\theta _0}\) . The set of strand-symmetric matrices is closed, so the assumption that each such \(P^{uv}\)  is strand-symmetric must be satisfied for all \(\theta \in \overline{\Theta }\) . Finally, if compositional asymmetry is violated at any internal node for a \(\theta \in \overline{\Theta }\) , then for at least one pair of terminal nodes \(a\)  and \(b\) , the eigendecomposition of \(S\left(J^{ab}\right)^{-1}SJ^{ab}\)  will have at least one repeated eigenvalue. That is, if \(\mathcal {P}_\theta =\mathcal {P}_{\theta _0}\)  for some \(\theta _0\in \Theta \) , then for \(\theta \)  we must have that all internal nodes satisfy compositional asymmetry.
